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1930s as part of his research of the foundations of mathematics. Lambda calculus consists of constructing lambda terms and performing reduction operations on them. Applying a function to an argument. M and N are lambda terms. Parentheses can be dropped if the expression is unambiguous. For some applications, terms for logical and mathematical constants and operations may be included. In typed lambda calculus, functions can be applied only if they are capable of accepting the given input’s “type” of data.

Subsequently, in 1936 Church isolated and published just the portion relevant to computation, what is now called the untyped lambda calculus. Until the 1960s when its relation to programming languages was clarified, the λ-calculus was only a formalism. The λ-calculus incorporates two simplifications that make this semantics simple. The first simplification is that the λ-calculus treats functions “anonymously”, without giving them explicit names. The second simplification is that the λ-calculus only uses functions of a single input. As described above, all functions in the lambda calculus are anonymous functions, having no names. They only accept one input variable, with currying used to implement functions with several variables.